
where step 1 follows from Assumption REF , step 2 comes
from similar reasoning as equation (REF ), and the last
step comes from \(\eta \le \frac{1}{L}\)  and \(\eta \le 1\)  and definition
of \(D_{3}\) . Therefore, from [1]} Lemma 3, the
time derivative of KL divergence along LMC is bounded by
\(\frac{d}{dt}H\left(p_{k,t}|\pi \right) & \le -\frac{3}{4}I\left(p_{k,t}|\pi \right)+\mathbb {E}_{p_{kt}}\left[\left\Vert \nabla U(x_{k,t})-\nabla U(x_{k})\right\Vert ^{2}\right]\\& \le -\frac{3}{4}I(p_{k}|\pi )+\frac{20N^{3}}{\gamma }L^{6}\eta ^{2\alpha }H(p_{k}|\pi )+D_{3}\eta ^{\alpha }\\& \le -\mathrm {\frac{3\gamma }{2}}H(p_{k,t}|\pi )+\frac{20N^{3}}{\gamma }L^{6}\eta ^{2\alpha }H(p_{k}|\pi )+D_{3}\eta ^{\alpha },\) 
